功能梯度压电材料板的有限元解

Finite Element Solution of Functionally Graded Piezoelectric Plates

本文利用变分原理和功能梯度压电材料的本构关系、几何关系、板的边界条件等,推导出功能梯度板的有限元方程。其中考虑了横向剪切变形的影响,采用了板变形问题的Mindlin假设,板内电势设为ψ=ψ0(x,y)+ ψ1(x,y)z+ψ2(x,y)z2+ψ1(x,y)g(z),并假设材料的力学和电学常数均沿板厚度z方向按同一函数规律K =K0f(z)变化,其中f(z)为任意的函数形式。为了验证本文方法的正确性,以功能梯度压电材料正方形板为例,使板所受的机械荷载和电荷载以及函数f(z)的形式与参考文献中所给出的相同,利用本文中提出有限元法计算了功能梯度板的电势和位移,所得结果与参考文献中的几乎一致。最后用此法计算四边简支．接地,线性梯度的PZT-4正方形板受均布荷载作用下的挠度和电势分布。

Using variational principle, constitutive relations and geometrical relations of functionally graded piezoelectric material, and the boundary conditions of the plates, the finite element equations were deducted. Considering the effect of transverse shear, the Reissner-Mindlin assumption for plates was employed. The electric potential of the plate was supposed to be Ф=ψ0（x,y）＋ψ1（x,y）z＋ψ2（x,y）z^2＋ψ1（x,y）g（z）. The mechanical and electrical constants of the material were supposed to vary according to K = K^0f（z） along the thickness, where .f（z） was an arbitrary function. A square plate was analyzed by means of this method to verify its validity. The results were compared with those given in some literatures. It is found that the results were almost the same as those in the literatures. Finally, this method was used to compute the deflections and electric potentials of a square plate made of PZT-4 with linear gradience of the material properties in the z-direction, under uniformly distributed load. The plate was simply supported and grounded on its four edges. Results of deflection and electric potential were provided on the symmetric axis.